# Why do drums and guitars sound differently?

This is a post from a series in my project.

What’s in a drum? How does it move? Does that affect the way it sounds? To understand these questions and how to begin to answer them, you should probably see how a drum head moves in slow motion. There are complicated waves that form on the surface of the drum head:

If you want to play with an interactive “drum,” go here.

Any partial differential equations (PDE) textbook will mention a drum head. Richard Haberman’s textbook presents the problem nicely. The drum head is the canonical example of a vibrating circular membrane. If you haven’t taken a course in PDEs, the book can be confusing. This post is supposed to be a general overview of a small section of the textbook, where Haberman talks about drum heads.

The model below explores the nature of sound waves on a finite interval in a 2-D space. Actual recordings of frequencies of a snare drum can be compared to the model and used to reconstruct the drum head at the time it was struck. The top five dominant frequencies of the recording represent the composition of the overall signal. The amplitudes corresponding to these frequencies are parameters in the computer representation of a the drum head. Basically, you can imagine that you’re blindfolded in the same room as a snare drum. You have a recording device to record the sound of the drum as someone strikes it, but you can’t look at it. All you can do is listen. After you finish recording, how can you use the data you have to reconstruct a picture of the drum at the time it was struck?

There are certain conditions and parameters that are important to this model.

• The boundary conditions will assume no movement along the fixed edge of the drum.
• This model will not assume radial symmetry, and approximates the solution to the two dimensional wave equation for the general case.
• Solving the general case requires special attention to $\theta$, the angle, in addition to $r$, the spatial displacement, and $t$, time, whereas solving the radially symmetric case assumes that the wave equation does not depend on $\theta$.
• Initial conditions will assume the drum has no initial velocity.
• Bessel functions of the first kind $J_n$ and their zeros are important components of the solution.

After I present the model of the drum, I’ll compare the differences in the harmonics and frequencies of the drum as opposed to those of a stretched string on a guitar. The model is based on the wave equation adapted for a circular membrane. The wave equation is an appropriate model for instruments because instruments use the fact that sound travels in waves to create music. Waves through guitar strings travel with little horizontal displacement, yet a drummer creates waves by displacing a stretched membrane, which no longer depends only on only vertical motion and yields a more complex problem.

# The wave equation as a model for the drum head

${{\partial}_t}^2 u = {c^2}{\nabla ^2}u = c^2[ \frac{1}{r} {\partial}_r(r{\partial}_r u) + \frac{1}{r^2}{\partial_{\theta}}^2 u]$

At a first glance, this model looks intimidating, but I assure you, it is not beyond your reach! Let me break it down:

• There is only one function in this mess. It is $u(r, \theta, t)$. In other words, the displacement of the membrane depends on time, the radius of the membrane, and an angular measurement that ranges from $[0, 2\pi]$ since we are on a circle.
• $c$ is a constant. Don’t worry about it.
• $\partial_t, \partial_r, \partial_\theta$ represent partial derivatives of $u(r,\theta,t)$. Think of it like you are seeing how the function changes when you hold 2 parameters constant and vary only one of them.
• $r$ is the spatial displacement, which can be anything between $[0,R]$. $R$ is the actual radius of a snare drum, which is something like $R = 15$ cm.

So now what do you do with this model? Answer: get an expression for $u(r,\theta,t)$. Using methods to solve this PDE (separation of variables), you can derive an equation of the same form as Bessel’s differential equation. As it turns out, Bessel and one of the Bernoullis did some of the leg work already (they derived these solutions back in the 1800s). We can just use their results – why reinvent the wheel?

# Results

This is the expression we’ve been looking for:

$u(r,\theta,t) = \sum_{i=1}^{\infty} \widehat{C}_{0,i}J_0(k_{0,i}r)\cos(k_{0,i}ct) + \sum_{n=1}^{\infty} \sum_{i=1}^{\infty} \widehat{C}_{n,i} \sin(n(\theta + \vartheta_n)) J_n(k_{n,i}r)\cos(k_{n,i}ct)$

Don’t sweat the details in this expression. What you should get out of it is that $u(r, \theta,t)$ is going to be a sum of a bunch of different terms. If we had infinitely many terms, our model of $u$ would be perfect. But, it’s the real world. And in the real world, you don’t get infinitely many terms. I got five of ’em.

You can take a Fast Fourier Transform of this signal data to get data that is in terms of amplitude and frequency. Essentially what I’m looking for is the top five dominant signals from this recording, but I can’t find them when the data is smooshed together like in this plot. The Fast Fourier Transform makes my life easier because it shows me the relationship between amplitude and frequency:

There are four obvious frequencies here, but the last one seems like it will be one of the really tiny bumps in the signal. Oh well, such is life.

# Reconstruction

1. Pick 5 dominant frequencies.
2. Plug them into the expression we derived for $u(r,\theta,t)$
3. Add up all of those numbers
4. Use your favorite software to generate plots that look like the ones below!

The images below are sort of lopsided. That makes me think that the drum was struck a touch off center. If you can hit the drum exactly in the middle, you would excite the most modes of the membrane, which would lead to the widest range of sounds you can hear.

There you have it! That’s how you reconstruct a drum head based on a recording of what it sounded like! Hooray! But the question still remains: why do drums and guitars sound different? This is the expression for the movement of a guitar string. Note that it has one less parameter than the expression for the drum head:

$u(x,t) = \sum_{n=1}^{\infty} \sin(\frac{n\pi x}{L})*[A_n \sin(\frac{cn\pi t}{L}) + B_n \cos(\frac{cn\pi t}{L})]$

You should know that the zeros of the spatial component of this string model are integer multiples of the others due to the periodic nature of sine and cosine. As a result, the natural frequencies of each string are in arithmetic progression so that when they are plucked, the overtone frequencies are harmonics of the fundamental frequency. Conversely, in the drum, the $u(r,\theta,t)$ is represented by a Bessel function instead of a trigonometric function. The zeros of the Bessel function are not integer multiples of each other, so the overtones are not harmonics of the fundamental frequency. The higher natural frequencies of a drum are random compared to the dominant frequency, since the Bessel function’s zeros are not integer multiples of each other. The most prominent sound is the dominant frequency of the drum, which corresponds to the largest amplitude spike or the first zero of the $J_0$ Bessel function. When combined with the dominant frequency, the higher frequencies (corresponding to the other zeros of the Bessel function), give the drum a “muffled” sound rather than a guitar-like “twang.”